An extremal nonnegative sine polynomial

摘要

For any positive integer n, the sine polynomials that are nonnegative in [0, pi] and which have the maximal derivative at the origin are determined in an explicit form. Associated cosine polynomials K-n(theta) are constructed in such a way that {K-n(theta)} is a surnmability kernel. Thus, for each p, 1 less than or equal to p less than or equal to infinity and for any 2pi-periodic function f is an element of L-p[-pi ,pi], the sequence of convolutions K-n * f is proved to converge to f in L-p [-pi, pi]. The pointwise and almost everywhere convergences are also consequences of our construction. [References: 16]